Convexity with Respect to a Linear Group

Abstract

In the preceding chapters we have seen that convex functions, subharmonic functions and plurisubharmonic functions are invariant under respectively the full linear group, the orthogonal group and the full complex linear group. We have also seen that the set of subharmonic functions invariant under the full (complex) linear group consists of convex (plurisubharmonic functions). In this chapter we shall consider the spaces of functions attached similarly to other subgroups of the linear group. These may be relevant in connection with the convexity with respect to certain differential operators as discussed in Chapter VI.